Digital Terrain Modeling: Principles and Methodology - Chapter 2 pot

2.2 NUMERIC TERRAIN DESCRIPTORS The complexity of a terrain surface may be described by the concepts of roughness and irregularity and characterized by different numerical parameters.. 2

Terrain Descriptors and Sampling Strategies

To model a piece of terrain surface, first a set of data points needs to be acquired from the surface Indeed, data acquisition is the primary (and perhaps the single most important) stage in digital terrain modeling For this, two stages are distinguished, that

is, sampling and measurement Sampling refers to the selection of the location while measurement determines the coordinates of the location Sampling will be discussed

in this chapter while measurement methods will be discussed in the next chapter Three important issues related to acquired DTM source (or raw) data are density, accuracy, and distribution The accuracy is related to measurements The optimum density and distribution are closely related to the characteristics of the terrain surface For example, if a terrain is a plane, then three points on any location will be sufficient This is not a realistic assumption and, therefore, an analysis of the terrain surface precedes the discussion of sampling strategies in this chapter

2.1 GENERAL (QUALITATIVE) TERRAIN DESCRIPTORS

In general, two basic types of descriptors may be distinguished:

1 qualitative descriptors, which are expressed in general terms, so that they are referred to as general descriptors

2 quantitative descriptors, which are those specified by numeric descriptors.

In this section, a brief discussion of general descriptors is given and numeric descriptors are described in the next section As discussed inChapter 1, different groups of people are concerned with different attributes of the terrain surface There-fore, a variety of general descriptors can be found based on these different interests However, some of them are irrelevant to the concern of digital terrain modeling Indeed, those that indicate the roughness and the coverage of terrain surface are more

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important in the context of terrain surface modeling The following are some of these descriptors:

1 Descriptors based on terrain surface cover: Vegetation, water, desert, dry soil,

snow, artificial or man-made features (e.g., roads, buildings, airports, etc.), and so on

2 Descriptors based on genesis of landforms: Two such forms have been distinguished

(Demek 1972), each of which has its own special characteristics —

• endogenetic forms: formed by internal forces, including neotectonic forms,

volcanic forms, and those forms resulting from deposition of hot springs

• exogenetic forms: formed by external forces, including denudation forms,

fluvial forms, karst forms, glacial forms, marine forms, and so on

3 Descriptors based on physiography: Generalized regions according to the structure

and characteristics of its landforms, each of which is kept as homogenous as possible and has dominant characteristics, for example, high mountains, high plateau, mountains, low mountains, hills, plateau, etc

4 Descriptors based on other classifications.

Those descriptors are so broad that they can only provide the user with some very general information about a particular landscape and thus they can only be used for general planning but not for project design To design a particular project, more precise numeric descriptors are essential

2.2 NUMERIC TERRAIN DESCRIPTORS

The complexity of a terrain surface may be described by the concepts of roughness and irregularity and characterized by different numerical parameters

2.2.1 Frequency Spectrum

A surface can be transformed from the space domain to the frequency domain by means of a Fourier transformation The terrain surface in its frequency domain is characterized by the frequency spectrum The estimation of such a spectrum from equally spaced discrete (profile) data has been discussed by Frederiksen et al (1978) The spectrum can be approximated by the following expression:

whereF denotes the frequency at which the spectrum magnitude is S(F ) and E and

a are constants (i.e., characteristic parameters), which are two statistics expressing

the complexity of the terrain surface (or profiles) over all of the area Thus, they can

be considered as parameters to provide more detailed information about the terrain surface, although still general in some sense

Different values for E and a can be obtained from different types of terrain

surfaces According to the study carried out by Frederiksen (1981), if the parametera

is greater than 2, the landscape is hilly with a smooth surface, and if the value of

a is smaller than 2, it indicates a flat landscape with a rough surface since the surface

contains large variations with high frequency (short wavelength) The value of a

provides us with general topographic information

2.2.2 Fractal Dimension

Fractal dimension is another statistical parameter which can be used to characterize the complexity of a curve or a surface The discussion will start with the concept of effective dimension

It is well known that in Euclidean geometry, a curve has a dimension of 1 and

a surface has a dimension of 2 regardless of its complexity However, in reality,

a very irregular curve is much longer than a straight line between the same points, and a complex surface has a much larger area than a plane over the same area In the extreme, if a line is so irregular that it fills a plane fully, then it becomes a plane, thus having a dimension of 2 Similarly, a surface could have a dimension of 3

In fractal geometry, which was introduced by Mandelbrot (1981), the dimension-ality of an object is defined by necessity (i.e., practical need), leading to the so-called effective dimension This can be explained by taking the example of the shape of the Earth’s surface when viewed from different distances

1 If it is viewed from an infinite distance, the Earth appears as a point, thus having a dimension of 0

2 If it is viewed from a position on the Moon, it appears to be a small ball, thus having

a dimension of 3

3 If the viewer comes nearer, for example, to a distance above the Earth’s surface of about 830 km (the altitude of the SPOT satellite’s orbit), the height information is extractable but not in detail Thus, in general terms, the observer can see a mainly smooth surface with a dimension of nearly 2

4 If the Earth’s surface is viewed on the ground, then the roughness of the surface can be seen clearly, thus the effective dimension of the surface should be greater than 2

In fractal geometry, the effective dimension could be a fraction, leading to the

jargon fractal dimension or fractal For example, the fractal dimension of a curve

changes between 1 and 2, and that of a surface between 2 and 3 The fractal dimension

is calculated as follows:

wherer is the scale of measurement (a principal unit), L is the length of

measure-ment,C is a constant, and D is interpreted as the fractal dimension of the curve line.

When measuring a fractal dimension of curve surface,r becomes the principal unit of

surface used for measurement and the resultant area isA instead of L; the expression

becomes

Figure 2.1 shows an example of Koch line with a fractal dimension of 1.26 The process of curve generation is as follows: (a) draw a line with its length as a unit;

(b)

(c)

(b) Divided into three line segments and mid-segment split into two (c) Process repeated.

left two lines are 0 as the radius is infinite while the line on the right side has large curvatures as the radiuses are small.

(b) divide the line into three segments; (c) the middle segment will be replaced by two polylines with length equal to13unit The same procedure is repeatedly applied to all line segments As a result, the line will become more and more complex, resulting in

a fractal dimension of 1.26

From the discussion above, it can be concluded that a fractal dimension approach-ing 3 indicates a very complex and probably rough surface, while a simple (near planar) surface has a fractal dimension value near 2

2.2.3 Curvature

The terrain surface can be synthesized by combing terrain form elements, defined

as relief unit of homogenous plan and profile curvatures (seeChapter 13for more details) Suppose a profile can be expressed asy = f (x), then the curvature at position

x can be computed as follows:

c = d2y/dx2

In this formula, curvaturec is inversely proportional to the radius of the curve (R),

that is, a large curvature is associated with a small radius (Figure 2.2) Thus, intuitively,

it can be seen that the larger the curvature, the rougher is the surface Therefore, curvatures can also be used as a measure for the roughness of the terrain This criterion has already been used for terrain analysis (e.g., Dikau 1989)

This is a comparatively useful method for planning DTM sampling strategies However, a rather large volume of data (that of a DTM) needs to be available to allow the curvature values to be derived — which leads to a chicken-and-egg situation

at the stage

2.2.4 Covariance and Auto-Correlation

The degree of similarity between pairs of surface points can be described by a cor-relation function This may take many forms like covariance or an auto-corcor-relation function The auto-correlation function is described as follows:

whereR(d) is the correlation coefficient of all the points with horizontal interval d,

Cov(d) is the covariance of all the points with horizontal interval d, and V is the

variance calculated from all the (N) points The mathematical functions are as follows:

V =

N

i=1 (Z i − M)2

Cov(d) =

N

i=1 (Z i − M)(Z i+d − M)

whereZ i is the height of pointi, Z i+d is the elevation of the point with an interval

ofd from point i, M is the average height value of all the points, and N is the total

number of points

When the value of d changes, Cov(d) and R(d) will also change because the

height difference of two points with differentd values is different Covariance and

auto-correlation values can be plotted against the distance between pairs of data points.Figure 2.3is an example of auto-correlations varying withd In general, if the

value of d increases, the values of Cov(d) and R(d) will decrease The curve is

usually described (Kubik and Botman 1976) by the exponential function:

and the Gaussian model:

Cov(d) = V × e −2d2/c2

(2.8) wherec is the parameter indicating the correlation distance at which the value of

covariance approaches 0 Therefore, the smaller the value ofc, the less similar are

the surface points

The value of similarity is also an indicator of the complexity of the terrain surface The relationship between them is that the smaller the similarity over the same given distance, the more complex is the terrain surface

2.2.5 Semivariogram

The variogram is another parameter used to describe the similarity of a DTM surface, similar to (auto-)covariance The expression for its computation is as follows:

2γ (d) =

N

i=1 (Z i − Z i+d )2

d R(d)

0

1

A B

between points from 1 to 0.

whereγ (d) is called the semivariogram Similar to covariance, the value of γ (d) will

vary with distance But the change in direction is opposite to the case of covariance That is,γ (d) will increase with an increase in the value of d The values of γ (d) can

also be plotted againstd, resulting a curved line Such a curve can be approximated

by an exponential function as follows:

whereA and b are two constants, i.e the two parameters for the description of terrain

roughness A largerb indicates a smother terrain surface When b is approaching zero,

the terrain is very rough Some examples of semivariograms are given inFigure 8.6

Indeed, Frederiksen et al (1983, 1986) used the semivariogram to describe

ter-rain roughness in digital terter-rain modeling They also tried to connect this variable

to the covariance used by Kubik and Botman (1976)

2.3 TERRAIN ROUGHNESS VECTOR: SLOPE, RELIEF,

AND WAVELENGTH

The numerical descriptors discussed in Section 2.2 are essentially statistical They are computed from a sample of terrain points from the project area Usually, some profiles are used as the sample and then a parameter is calculated from these profiles However, there are some problems associated with this approach One of these is that the parameters calculated from the selected profiles can be different from those derived from the whole surface If one tries to compute these for the whole surface, then a sample from the whole surface is necessary In this case, the original purpose of having

a terrain descriptor for project planning and design is lost For these reasons, Li (1990) recommended slope and wavelength as the main descriptors for DTM purposes

2.3.1 Slope, Relief, and Wavelength as a Roughness Vector

The parameters for roughness or complexity of a terrain surface used in geomorphol-ogy have also been reviewed by Mark (1975) It was found that roughness cannot be

completely defined by any single parameter, but must be a roughness vector or a set

of parameters

H = amplitude

Slope angle of P

W H

W= wavelength

and (b) simplified diagram.

In this set of parameters, relief is used to describe the vertical dimension (or amplitude of the topography), while the terms grain and texture (the longest

and shortest significant wavelengths) are used to describe the horizontal variations (in terms of the frequency of change) The parameters for these two dimensions are connected by slope Thus, relief, wavelength, and slope are the roughness parameters The relationship between them can be illustrated in Figure 2.4 It can clearly be

seen that the slope angle at a point on the wave varies from position to position.

The following mathematical equation may be used as an approximate expression

of their relationship (for a more rigorous definition, seeChapter 13):

tanα = H

W/2 =

2H

whereα denotes the average value of the slope angle, H is the local relief value

(or the amplitude), andW is the so-called wavelength It is clear that if two of them

are known, then the third can be computed from Equation (2.11) For the reasons

to be discussed in the next section, slope and wavelength together are recommended

as the terrain roughness vector for DTM purposes

2.3.2 The Adequacy of the Terrain Roughness

Vector for DTM Purposes

From both the theoretical and the practical points of view, slope, altitude, and wavelength are the important parameters for terrain description

In geomorphology, Evan (1981) states

a useful description of the landform at any point is given by altitude and the surface derivatives, i.e slope and convexity (curvature) Slope is defined by a plane tangent

to the surface at a given point and is completely specified by the two components: gradient (vertical component) and aspect (plane component) Gradient is essentially

the first vertical derivative of the altitude surface while aspect is the first horizontal derivative

Further, land surface properties are specified by convexity (positive and negative convexity — concavity) These are the changes in gradient at a point (in profile)

and the aspect (in the plane tangential to the contour passing through the point).

In other words, they are second derivatives These five attributes (altitude, gradient, aspect, profile convexity, and plane convexity) are the main elements used to describe terrain surfaces Among them, slope, comprising of both gradient and aspect, is the fundamental attribute

Gradient should be measured at the steepest direction However, when taking the gradient of a profile or in a specific direction, it is actually the vector of the gradient and aspect that is obtained and used Therefore, the term slope or slope angle is used

in this context to refer to the gradient in any specific direction

The importance of slope has also been realized by others As quoted by Evans (1972), Strahler (1956) pointed out that “slope is perhaps the most important aspect

of surface form, since surfaces may be formed completely from slope angles .”

Slope is the first derivative of altitude on the terrain surface It shows the rate of change in height of the terrain over distance

From the practical point of view, using slope (and relief ) as the main terrain descriptor for DTM purposes can be justified for the following reasons:

1 Traditionally, slope has been recognized as very important and used in surveying and mapping For example, map specifications for contours are given in terms of slope angle all over the world

2 In the determination of vertical contour intervals (CIs) for topographic maps, slope and relief (height range) are the two main parameters considered For example, Table 2.1 is a classification system adopted by the Chinese State Bureau of Sur-veying and Mapping (SBSM) in its specifications for 1:50,000 topographic maps

3 In DTM practice, many researchers (e.g., Ackermann 1979; Ley 1986; Li 1990, 1993b) have noted the high correlation between DTM errors and the mean slope angle of the region

2.3.3 Estimation of Slope

To use slope together with wavelength or relief to describe terrain, two problems

related to the estimation of its values need to be considered, that is, availability and

variability.

By availability we mean that slope values should be available or estimated before

sampling takes place, to assist in the determination of sampling intervals If a DTM exists in an area, then the slope values for DTM points can be computed and the average can be used as the representative (Zhu et al 1999) Otherwise, slope may be estimated from a stereo model formed by a pair of aerial photographs with overlap (seeChapter 3)or from contour maps The method proposed by Wentworth (1930)

is still widely used to estimate the average slope of an area from the contour maps

The average slope value (α) of a homogeneous are can be estimated as follows:

α = arctan



H × !L

A



(2.12)

where H is the contour interval, !L is the total length of contours in the area and A

is the size of the area If there is no contour map for such an area, then the slope may

be estimated from an aerial photograph Some of the methods that are available for measurement of slope from aerial photographs have been reviewed by Turner (1997)

By variability we mean that slope values may vary from place to place so that

the slope estimate that is representative for one area may not be suitable for another

In this case, average values may be used as suggested by Ley (1986) If slope varies too greatly in an area, then the area should be divided into smaller parts for slope estimation Different sampling strategies could be applied to each area

2.4 THEORETICAL BASIS FOR SURFACE SAMPLING

After estimating slope and relief (height range), the wavelengths of terrain variation can be computed These parameters are used to determine the sampling strategy and intervals for data acquisition First, some theories related to surface sampling are discussed

2.4.1 Theoretical Background for Sampling

From the theoretical point of view, a point on the terrain surface is 0-D, thus without size, while a terrain surface comprises an infinite number of points If full information about the geometry of a terrain surface is required, it is necessary to measure an infin-ite number of points This means that it is impossible to obtain full information about the terrain surface However, in practice, a point measured on a surface represents the height over an area of a certain size; therefore, it is possible to use a set of finite points to represent the surface Indeed, in most cases, full or complete information about the terrain surface is not required for a specific DTM project, so it is necessary only to measure enough data points to represent the surface to the required degree of accuracy and fidelity

The problem a DTM specialist is concerned with is how to adequately represent the terrain surface by a limited number of elevation points, that is, what sampling interval

to use with a known surface (or profile) The fundamental sampling theorem that is being widely used in mathematics, statistics, engineering, and other related disciplines can be used as the theoretical basis The sampling theorem can be stated as follows:

If a functiong(x) is sampled at an interval of d, then the variations at frequencies

higher than 1/(2d) cannot be reconstructed from the sampled data.

That is, when sampling takes two samples (i.e., points) from each period of waves with the highest frequency in the functiong(x), the original g(x) can be completely

reconstructed with the sampled data In the case of terrain modeling, if a terrain profile

is long enough to be representative of the local terrain, it can then be represented by the

Figure 2.5 The relationship between the least sampling interval and the highest functional

frequency Left: sampling interval is less than half the functional frequency

so that full reconstruction is possible; right: sampling interval is larger than half the functional frequency so that information about the function is lost.

sum of its sine and cosine waves If it is assumed that the number of terms in this sum

is finite, there is, therefore, a maximum frequency value,F , for this set of sinusoidal.

According to the sampling theorem, the terrain profile can be completely reconstruc-ted if the sampling interval along the profile is smaller than 1/(2F ) (see Figure 2.5,

left) Therefore, extending this idea to surfaces, the sampling theorem can also be used to determine the sampling interval between profiles to obtain adequate inform-ation about a terrain surface In contrast, if a terrain profile is sampled at an interval

ofd, then the terrain information with a wavelength less than 2d will be completely

lost (Figure 2.5, right) Therefore, as Peucker (1972) has pointed out, “a given regular grid of sampling points can depict only those variations of the data with wave lengths

of twice the sampling interval or more.”

2.4.2 Sampling from Different Points of View

Points on a terrain surface can be viewed in various ways from the differing view-points inherent in subjects such as statistics, geometry, topographic, science, etc Therefore, different sampling methods can be designed and evaluated according to each of these different viewpoints as follows (Li 1990):

1 statistics-based sampling

2 geometry-based sampling

3 feature-based sampling

From the statistical point of view, a terrain surface is a population (called

a sample space) and the sampling can be carried out either randomly or systematically The population can then be studied by the sampled data In random sampling, any sampled point is selected by a chance mechanism with known chance of selection The chance of selection may differ from point to point If the chance is equal for all sampled points, it is referred to as simple random sampling In systematic sampling, the points are selected in a specially designed way, each with a chance of 100% probability of being selected Other possible sampling strategies are stratified sampling and cluster sampling However, they are not suitable for terrain modeling and thus are omitted here

From the geometric point of view, a terrain surface can be represented by different geometric patterns, either regular or irregular in nature The regular pattern can be subdivided into 1-D or 2-D patterns If sampling is conducted with a regular pattern